含不連續(xù)系數(shù)的時(shí)滯微分方程奇攝動(dòng)邊值問(wèn)題

陽(yáng)廣志,謝 峰

(東華大學(xué) 應(yīng)用數(shù)學(xué)系,上海 201620)

含不連續(xù)系數(shù)的時(shí)滯微分方程奇攝動(dòng)邊值問(wèn)題

陽(yáng)廣志,謝 峰

(東華大學(xué) 應(yīng)用數(shù)學(xué)系,上海 201620)

研究一類含不連續(xù)系數(shù)的二階時(shí)滯微分方程的奇攝動(dòng)問(wèn)題.原問(wèn)題可看作左問(wèn)題與右問(wèn)題的藕合,利用邊界層函數(shù)法分別構(gòu)造左右問(wèn)題的漸近解從而得到其解的零階近似, 再利用縫接法得到整個(gè)區(qū)間上的解,最后運(yùn)用上下解引理證明解的存在性.

時(shí)滯;不連續(xù)系數(shù);奇攝動(dòng);上下解;縫接法

0 引 言

關(guān)于時(shí)滯微分方程的邊值問(wèn)題已受到廣泛而深入的討論[1-5], 但主要都集中在連續(xù)情形下的奇異攝動(dòng)時(shí)滯問(wèn)題[6-10], 對(duì)一些含不連續(xù)系數(shù)或不連續(xù)源項(xiàng)的奇攝動(dòng)時(shí)滯問(wèn)題研究相對(duì)較少. 近幾十年來(lái), 在物理、 化學(xué)、 生物以及環(huán)境學(xué)等許多領(lǐng)域的數(shù)學(xué)模型都可歸結(jié)為帶有不連續(xù)系數(shù)或不連續(xù)源項(xiàng)的奇攝動(dòng)時(shí)滯問(wèn)題,并備受研究者重視,但研究主要集中在數(shù)值方法方面[11-12].2015年,Subburayan[12]研究了一類具有不連續(xù)對(duì)流系數(shù)的二階線性奇攝動(dòng)時(shí)滯問(wèn)題，并結(jié)合文獻(xiàn)[13]給出了分段線性插值Shishkin網(wǎng)格的數(shù)值方法.

本文主要研究下述含不連續(xù)源項(xiàng)的時(shí)滯微分方程的奇攝動(dòng)邊值問(wèn)題

εx″(t)+f(t,x(t))x′(t)=g(t,x(t),x(t-τ)),t∈(0,τ)∪(τ,1),

(1)

x(t)=φ(t),t∈[-τ,0];x(1)=A.

(2)

其中

1 上下解引理

考慮如下非線性邊值問(wèn)題

x″(t)=f(t,x(t),x(t-τ),x′(t)),t∈(0,τ)∪(τ,1),

(3)

x(t)=φ(t),t∈[-τ,0];x(1)=A.

(4)

其中

引理1 假設(shè)(H1)f(t,x,y,z)于[0,τ)∪(τ,1]×R3上連續(xù),且對(duì)?r>0,存在[0,+∞)上的正值連續(xù)函數(shù)h(s)滿足

且當(dāng)t∈[0,1],|x|≤r,|y|≤r,|z|<∞時(shí)有

|f(t,x,y,z)|≤h(|z|).

(H2)f(t,x,y,z)關(guān)于y單調(diào)不增.

(H3)存在函數(shù)α(t),β(t)∈C[-τ,1]∩C2[0,τ)∪(τ,1],滿足

α(t)≤β(t),t∈[-τ,1];α′(τ-)≤α′(τ+),β′(τ-)≥β′(τ+),

α″(t)≥f(t,α(t),x(t-τ),α′(t)),t∈(0,τ)∪(τ,1),

β″(t)≤f(t,β(t),x(t-τ),β′(t)),t∈(0,τ)∪(τ,1)

則?φ(t)∈C[-τ,0],當(dāng)α(t)≤φ(t)≤β(t),t∈[-τ,0]，α(1)≤A≤β(1)時(shí),邊值問(wèn)題(3)～(4)有解x(t)∈C1[-τ,1]∩C2([0,τ)∪(τ,1]), 滿足

α(t)≤x(t)≤β(t),t∈[0,1].

(5)

其中

yβ(τ)=β(t-τ),yα(τ)=α(t-τ).

又令

有解.

下面先證明解x(t)滿足不等式(5).以下只證明x(t)≤β(t)(因?yàn)棣?t)≤x(t)可類似處理). 用反證法, 設(shè)存在一點(diǎn)t0∈[0,1]使得x(t0)>β(t0), 則由于x(0)=φ(0)≤β(0), x(1)=A≤β(1), 函數(shù)x(t)-β(t)必在(0,1)內(nèi)的某點(diǎn)ξ處取正的極大值, 從而x(ξ)>β(ξ), x′(ξ)=β′(ξ), x″(ξ)≤β″(ξ). 但依據(jù)(H2)和(H3), 當(dāng)ξ∈(0,τ)時(shí)，有

這與x″(ξ)-β″(ξ)≤0矛盾.

同理當(dāng)ξ∈(τ,1)時(shí)也推出矛盾.

當(dāng)ξ=τ時(shí),x(t)-β(t)在τ處取得正的極大值,從而有

類似可得x″(τ+)-β″(τ+)>0,這與x″(τ±)≤β″(τ±)矛盾.

再證|x′(t)|≤M(0≤t≤1),若此不等式不成立,即存在一點(diǎn)t1∈[0,1],使得|x′(t1)|>M.由中值定理,必存在ξ∈(0,1)使得x′(ξ)=x(1)-x(0),故|x′(ξ)|≤2r.又從x′(t)的連續(xù)性知,存在τ1,τ2∈[0,1],使得

|x′(τ1)|=2r,|x′(τ2)|=M,

且當(dāng)τ1

這與M的取法矛盾, 故|x′(t)|≤M(0≤t≤1). 即有|x(t)|≤M(0≤t≤1), 從而x(t)就是原問(wèn)題的解.

2 漸近解的構(gòu)造

由于原問(wèn)題在t=τ處間斷, 所以會(huì)產(chǎn)生內(nèi)部層, 因此可以把問(wèn)題(1)～(2)看成是以下兩個(gè)問(wèn)題的光滑連接.

左問(wèn)題PL:

εx″(t)+f1(t,x(t))x′(t)=g1(t,x(t),φ(t-τ)),t∈(0,τ),

(6)

x(0)=φ(0),xL(τ)=γ(ε).

(7)

右問(wèn)題PR:

εx″(t)+f2(t,x(t))x′(t)=g2(t,x(t),x(t-τ)),t∈(τ,1),

(8)

xR(τ)=γ(ε),x(1)=A.

(9)

其中γ(ε)是與ε有關(guān)的待定參數(shù).

現(xiàn)作如下假設(shè):

(ⅰ) 函數(shù)f1(t,x), g1(t,x,y)在[0,τ]×R3上連續(xù), f2(t,x), g2(t,x,y)在[τ,1]×R3上連續(xù), 且gi(i=1,2)關(guān)于y單調(diào)不增;

(ⅱ) 初值問(wèn)題f1(t,x(t))x′(t)=g1(t,x(t),φ(t-τ)), x(0)=φ(0)有解φ1(t)∈[0,τ];

(ⅲ) 初值問(wèn)題f2(t,x(t))x′(t)=g2(t,x(t),x(t-τ)), x(1)=A有解φ2(t)∈C2[0,1];

(ⅳ) f1(t,x)≤-σ1<0,(t,x)∈[0,τ]×R; f2(t,x)≥σ2>0,(t,x)∈[τ,1]×R.

首先考慮左問(wèn)題的漸近解構(gòu)造, 假設(shè)左問(wèn)題PL的形式漸近解表達(dá)式為

(10)

其中

(11)

VL(η)=V0L(η)+εV1L(η)+ε2V2L(η)+…,

(12)

γ(ε)=γ0+εγ1+ε2γ2+….

(13)

將式(10)代入式(6)并分離變量，再根據(jù)ε同次冪系數(shù)相等, 可得到一系列遞推等式. 為了簡(jiǎn)單起見(jiàn), 以下只考慮零階近似.

(14)

同時(shí)可得邊界層項(xiàng)V0L(η)滿足如下邊值問(wèn)題

(15)

(16)

由文獻(xiàn)[16]中引理1以及條件(ⅰ),(ⅱ),(ⅳ)可知, 邊值問(wèn)題(15)～(16)存在解V0L(η)且滿足指數(shù)估計(jì), 即存在正常數(shù)K1, 使得

|V0L(η)|≤K1exp(σ1η).

因此, 可得左問(wèn)題的零階近似漸近解形式為

xL(t)=φ1(t)+V0L(η),t∈[0,τ].

(17)

同左問(wèn)題, 可得右問(wèn)題的零階近似漸近解形式為

(18)

(19)

邊界層項(xiàng)V0R(η)滿足邊值問(wèn)題

(20)

(21)

由文獻(xiàn)[16]中引理2以及條件(ⅰ),(ⅲ),(ⅳ)可知, 邊值問(wèn)題(20)～(21)存在解V0R(η)且滿足指數(shù)估計(jì), 即存在正常數(shù)K2, 使得

|V0R(η)|≤K2exp(-σ2η).

為了使左問(wèn)題的解與右問(wèn)題的解在t=τ處光滑連接,須有

(22)

(17),(18)代入代(22)得

(23)

(24)

同理, 由式(20)和(21)可推出

(25)

將式(24), (25)代入式(23)得

(26)

易知式(26)的右邊是負(fù)值, 則γ0可以由式(26)決定.同理γi(i≥1)也可由此遞推得到.

因此, 可得問(wèn)題(1)～(2)的形式漸近解.

定理1 假設(shè)條件(ⅰ)～(ⅳ)成立, 則對(duì)于充分小的ε>0, 邊值問(wèn)題(1)～(3)有一個(gè)C1光滑解x(t)滿足

其中

3 定理的證明

現(xiàn)證明定理1,首先構(gòu)造合適的上下解.記

根據(jù)近似解的構(gòu)造過(guò)程和假設(shè)(ⅰ)知, 存在正常數(shù)M使得

|εx″(t)+f(t,x(t)x′(t)-g(t,x(t),x(t-τ)))|≤Mε2,t∈[0,1],

(27)

(29)

令

其中

容易驗(yàn)證, λ(t)是一個(gè)正值連續(xù)函數(shù)且具有如下性質(zhì)

(30)

(31)

再依據(jù)式(29)～(30), 有

α(t),β(t)∈C([0,1])，α(t)<β(t),t∈[0,1],

α′(τ-)≤α′(τ+),β′(τ-)≥β′(τ+)，α(0)<φ(0)<β(0),α(1)

下面驗(yàn)證不等式εα″(t)+f(t,α(t))α′(t)≥g(t,α(t),α(t-τ)).僅考慮區(qū)間(τ,1)上,在(0,τ)上可類似證明.根據(jù)式(27), (28)以及(30), (31)可得

其中0<θ1,θ2,θ3<1.

類似可得對(duì)充分小的ε>0,不等式

εβ″(t)+f(t,β(t))β′(t)≤g(t,β(t),β(t-τ)),t∈(0,τ)∪(τ,1)

成立.

以上證明了α(t),β(t)分別是問(wèn)題(1)～(3)的下解和上解. 由上下解引理可知, 問(wèn)題(1)～(2)存在解

x(t)∈C1[0,1]∩C2((0,τ)∪(τ,1)),

且?t∈[0,1]有α(t)≤x(t)≤β(t),證畢.

4 應(yīng)用舉例

考慮如下問(wèn)題

(32)

將問(wèn)題(32)看成是以下兩個(gè)問(wèn)題的光滑連接.

左問(wèn)題：

(33)

右問(wèn)題:

(34)

其中γ(ε)是與ε有關(guān)的待定參數(shù).可令

γ(ε)=γ0+εγ1+ε2γ2+…,

根據(jù)前面的構(gòu)造方法, 可得左右問(wèn)題的退化解分別為φ(x)∈[0,1],ψ(x)∈[0,2], 且

再由式(15)～(16)以及(20)～(21)可求出左右問(wèn)題邊界層函數(shù)的零階近似V0L(η),V0R(η)分別為

從而問(wèn)題(32)的零階近似解可表示為

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編輯、校對(duì)：師 瑯

Singularly perturbed boundary value problems of differential equations with delay and discontinuous coefficients

YANGGuangzhi,XIEFeng

(Department of Applied Mathematics, Donghua University, Shanghai 201620, China)

A class of singularly perturbed problems of second-order delay differential equations with discontinuous coefficients are studied. The original problem can be viewed as the coupling of the left and right problem. Asymptotic solutions of the left and right problem are constructed by using the method of boundary function respectively, so that the solution of zero order approximation is obtained. To make the solution set up on the whole interval, the sewing method is used. At last, the existence of solution are proved by the theorem of lower and upper solutions.

delay; discontinuity coefficients; singular perturbation; lower and upper solution；sewing method

1006-8341(2016)04-0435-08

10.13338/j.issn.1006-8341.2016.04.004

2016-05-28

上海市自然科學(xué)基金資助項(xiàng)目(15ZR1400800)

謝峰(1976—),男,安徽省宿州市人，東華大學(xué)教授,研究方向?yàn)槌Ｎ⒎址匠膛c動(dòng)力系統(tǒng)，奇異攝動(dòng)理論及其應(yīng)用.E-mail:fxie@dhu.edu.cn

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YANG Guangzhi XIE Feng.Singularly perturbed boundary value problems of differential equations with delay and discontinuous coefficients[J].Basic Sciences Journal of Textile Universities,2016,29(4):435-442.

O 175.14

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